The natural oscillations of distributed inhomogeneous systems described by generalized boundary value problems

The problem of the constructive determination of the natural frequencies and modes of oscillations of distributed systems with substantially varying parameters is investigated. Unlike the classical case, the self-adjoint boundary-value problem allows of an arbitrary non-linear dependence of the coefficients of the equation on a numerical parameter, the eigenvalues of which are required to be obtained. An original numerical-analytic method is developed for a highly accurate construction of the desired solution. The computational efficiency of the algorithm, which possesses the property of accelerated (quadratic) convergence, is illustrated by the calculation of model examples. The approach can be extended to other classes of generalized problems of determining the critical values of the parameters and the forms corresponding to them, in particular, to the problem of the loss of stability of elastic systems with variable stiffnesses and inertial and force characteristics. A highly accurate solution of the classical Prandtl problem of determining the critical force which leads to lateral buckling of a long homogeneous cantilever beam is constructed, taking its weight into account.